Multiobjective Optimization for Robust Holonomic Quantum Gates

TL;DR

The paper addresses the challenge of designing high-fidelity holonomic quantum gates that are robust to multiple, often conflicting error sources. It develops a universal multiobjective optimization framework that combines Pareto optimization with an entropy-weight method (EWM) and applies it to Rydberg-based nonadiabatic holonomic gates, yielding ADR and SR pulses that simultaneously mitigate amplitude and detuning errors and reduce decoherence by shortening Rydberg dwell time. Compared with traditional one-objective optimization, the multiobjective approach delivers higher-fidelity gates for both single-qubit X and two-qubit CNOT operations under realistic noise, demonstrating the practical viability of fault-tolerant NHQC. The framework is generalizable to other gate protocols, enabling principled trade-offs among multiple error channels and contributing to systematic error budgeting in quantum gate engineering.

Abstract

The practical implementation of high-fidelity quantum gates faces significant challenges in simultaneously mitigating multiple operational errors arising from distinct physical mechanisms. These errors often span orders of magnitude in severity, and their respective suppression strategies may inherently conflict. In this work, we develop a universal multiobjective optimization framework for quantum gate design by integrating Pareto optimal solutions with an entropy-weight method. Using Rydberg-based nonadiabatic holonomic quantum gates (affected by amplitude errors, detuning errors, and Rydberg decoherence) as a testbed, we theoretically demonstrate the superiority of our algorithm. The optimized gates exhibit enhanced fidelity and robustness compared to those derived from one-objective optimization strategies. Furthermore, this framework is readily adaptable to other quantum gate protocols and provides a robust foundation for advancing fault-tolerant quantum computing.

Figures (7)

• Figure 1: Basic idea of single-qubit nonadiabatic holonomic quantum gates. (a) Energy levels for a three-level atom, where the qubits are encoded in ground hyperfine states ${\vert 0 \rangle},\vert 1\rangle$, and the Rydberg state $\vert r \rangle$ acts as an auxiliary level that spontaneously decays to the ground states. The ground-Rydberg transitions between $|r\rangle$ and $|0\rangle$ (or $|1\rangle$) are realized by off-resonant two-photon laser fields with a common detuning $\Delta(t)$. (b) Effective two-level system in the $\{|b\rangle,|r\rangle\}$ subspace where the dark state $|d\rangle$ is decoupled. Inset: Illustration of the second dark path $|\mu_2(t)\rangle$ evolution with a close loop on the Bloch sphere of $\{|b\rangle,|r\rangle\}$ subspace, representing the realization of single-qubit gate operation based on the TO pulse.
• Figure 2: Gate performance based on the typical one-objective optimization with respect to (a1) the $\epsilon$ error, (a2) the $\eta$ error and (a3) the $\kappa$ error.
• Figure 3: Dependence of the gate infidelity $1-\mathcal{F}$ of different pulses in the absence of decoherence $\kappa=0$ on (a) the amplitude error $\epsilon$ with $\eta=0$ and (b) the detuning error $\eta$ with $\epsilon=0$.
• Figure 4: (a) PF formed by sufficient Pareto points ($n=100$) for (a1) two objectives ($\mathcal{J}_{dr},\mathcal{J}_{ar}$) and (a2) three objectives ($\mathcal{J}_{dr},\mathcal{J}_{ar},\mathcal{J}_{dcr}$). (b) The ideal population dynamics in the absence of any error $(\epsilon,\eta,\kappa)=0$ with (b1) the ADR pulse and (b2) the SR pulse.
• Figure 5: For a practical simulation, the gate infidelity as a function of both $\eta$ and $\epsilon$ in the presence of decoherence $\kappa/2\pi=3.0$ kHz using (a) the ADR pulse and (c) the SR pulse. The corresponding evolution path of $|\mu_2(t)\rangle$ for the $X$ gate on the Bloch sphere are shown in (b) and (d).